Activity Raoultian

The numerical value of the activity of j at the Henrian standard state is 1 on the Henrian activity scale, but y° on the Raoultian activity scale. 

An activity coefficient of a constituent in a system is a number (always dimensionless) which when multiplied by the ideal activity (the concentration) gives the real activity. To illustrate this let s examine a system of the type in Figure 11.9 in more detail (Figure 11.10). In this binary system, B shows positive deviation from Raoultian behavior, so the Raoultian activity coefficient will be greater than one. Solutes that obey Raoult s Law have a, = X, so the Raoultian activity coefficient 7h is defined as... 

Fig. 11.11. Raoultian activities of H2O and CO2 in the binary solution at 600°Cand 2 kb. Data from Bowers and Helgeson (1983). The curved lines are fit to the data with Margules equations, discussed in Chapter 15. The inset refers to a discussion of standard states in Chapter 12.

This leads to the easiest approach to understanding activities. The activity of a constituent is the ratio of the fugacity of that constituent to its fugacity in some other state, which we called a reference state. We then showed through consideration of the Lewis Fugacity Rule, which is an extension of Dalton s Law, that for ideal solutions of condensed phases, the activity of a constituent equals its mole fraction, if the reference state is the pure constituent at the same P and T. Deviations from ideal behaviour are then conveniently handled by introducing Henryan and Raoultian activity coefficients. 

Equation (12.10) can also be used for non-ideal gases as well as for liquid and solid solutions by introducing the Raoultian activity coefficient, thus... 

Fig. 12.1. One possible relationship between Henryan and Raoultian activities (essentially the same as Figure 11.10).

The physical meaning of this choice can be illustrated by returning first to Figure 11.10. There we have a solute B that has an activity of 0.5 at a mole fraction of 0.3, using our Raoultian activity scale based on fugacities. We can look at the same system from a molality point of view by letting component A be water (55.51 moles H2O/kg), so that ua = 55.51. At Xb = 0.3, B then has a molality (me) of about 23.8, far beyond the concentration at which most solid substances become saturated in water. In other words we cannot simply let Figure 11.10 represent an aqueous... 

For solutions covering a wide range of compositions, such as many solid and liquid solutions, this equation can be used by introducing another correction factor, the Raoultian activity coefficient, Thus... 

To illustrate the Raoultian activity-activity coefficient relationship we use activity coefficients defined by Equations (8.31), shown in Figure 8.2. In real systems these are measured quantities with associated uncertainties, and the shape of the activity curve may not fit any simple function. Figure 8.2 shows... 

Figure 8.3 Henry s law tangent (Figure 8.2) extended to intersect the Xb = 1 axis. Raoultian activities from Figure 8.2 are also shown. Henryan activities shown are Th-Xb-...

Figure 8.7 Henryan activity of sucrose, Raoultian activity of water, and osmotic coefficients in sucrose solutions at 25°C. The inset is an enlargement of the region up to 1 molal, and the circle shows the ideal one molal standard state.

An interesting application of regular solution theory is presented by Nesbitt (1984). He shows that activity coefficients for COj in aqueous NaCl solutions to quite high temperatures ( 500°C) and NaCl concentrations ( 6 m) can be fit very well by a slight modification of (10.98). As written, the activity coefficients in (10.98) are based on Raoultian activities that is, 7b 1 as Xg 1. Solubility studies on the other hand normally use Henryan coefficients, where 7b 1 as Wb 0, where is the molality of the solute. 

The rest of this chapter is in two parts. First, we consider how to calculate mole fractions in solid solutions assumed to have only long range order, and how to combine these mole fractions into Raoultian activities. Then we consider the determination of activity coefficients in (binary) solid solutions, and how regular solution and Margules equations are used to systematize these. 

We know that for Raoultian ideal solutions, O = x or activity is equal to mole fraction, so to calculate the Raoultian activity of solid solution components, all we have to do is calculate their mole fractions. If there is only one... 

After equilibrating the phases at T and P, the garnet is analyzed, and the ideal Raoultian activity of Ca3Al2Si30,2 is calculated from the thermodynamic mole reaction as in Equations (14.8), i.e.. 

A non-ideal (or actual) solution is one for which Eq.(6.22) does not hold good for at least one component. A correction factor > known as the Raoultian activity coefficient of component i, is introduced to Eq.(6.22) so that it may also be applied to non-ideal solutions. Thus... 

Vanadium melts at 1720°C (1993 K). The Raoultian activity coefficient of vanadium at infinite dilution in liquid iron at 1620°C (1893 K) is 0.068. Calculate the free energy change accompanying the transfer of the standard state from pure solid vanadium to the infinitely dilute, weight percent solution of vanadium in pure iron at 1620°C. 

The excess term in the chemical potential = /xa — Ma hence, RT In fA is obtained from AmG and is ° W(1 —xa). This term goes to zero when xa 1 (cf. Raoult s law). The activity coefficient just considered therefore, is referred to as the Raoultian activity coefficient. When xa 0 it is constant (cf. Hemy s law). The displacement of the constant W from the term to the /x° term makes possible another normalization method, which is advantageous for describing dilute states. The first normalization chosen above and characterized by 1a —1 for xa 1 is known as the Raoultian normalization, and the second possibility with fA 1 for xa 0 as the Henryan normalization. There are then two equivalent representations of /xA) namely as + RTlnxA -I- RTln fA or as -h RTlnxA + RTln fA- In the regular model /Xa — W and, hence, In fA = W(xa — 2xa)/RT while... 

Fig. 4.11 Heniyan and Raoultian activities. In the regular model (here W>0) both activity scales are proportional If = a/ %i = = exp(W/RT). Generally the scale is...

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